Real Analysis 2


This module builds on the level 1 module Real Analysis 1 and focuses on continuity, differentiability and integrability of real functions of one variable.


This module contains the following topics:

  • Limits of functions, continuity (ε-δ definition). Sums, products, compositions. Intermediate value theorem and extreme value theorem
  • Differentiable functions (sums, products, quotients). Differentiability implies continuity. Chain rule, inverse functions. Rolle's theorem, mean value theorem, l'Hôpital's rule. Higher derivatives. Taylor 's theorem. Contraction mapping theorem.
  • Theory of integration: upper and lower sums and integrals, the Riemann integral. Conditions for integrability (e.g., continuity implies integrability). Indefinite integration, and the fundamental theorem of calculus. Taylor series with integral remainder.

Selected Texts

Background Reading

  • J. M. Howie, Real Analysis, Springer (2001). (Available in paperback from UniS bookshop, Springer or
  • W.F. Trench, Introduction to Real Analysis, (2003, updated February 2010). Can be downloaded free via
  • Ethan D. Bloch, The Real Numbers and Real Analysis, Springer, 2010. DOI: 0.1007/978-0-387-72177-4.  Can be downloaded free via
  • Sudhir R. Ghorpade and Balmohan V. Limaye, A Course in Calculus and Real Analysis, Undergraduate Texts in Mathematics, 2006. DOI: 10.1007/0-387-36425-0.  Can be downloaded free via
  • J. Lewin, Mathematical Analysis, Cambridge University Press (2003).
  • P.E. Kopp, Analysis, Arnold Publishers, (1990).
  • S. Lang, Analysis I, Addison-Wesley (1968).
  • J.E. Snow and K.E. Weller, Exploratory Examples for Real Analysis, Cambridge University Press (2004).

Page Owner: mt0019
Page Created: Monday 26 November 2012 09:04:10 by mt0019
Last Modified: Thursday 31 January 2013 16:07:22 by mt0019
Expiry Date: Wednesday 26 February 2014 09:03:48
Assembly date: Fri Apr 08 20:52:32 BST 2016
Content ID: 94405
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